# What is the relative strength of each of the players in this game?

This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on their team. So far 17 people have turned up to these games. They are each represented by a different letter in the alphabet from A to Q. The players have different capabilities. The teams are usually 5 on 5 but it can vary if an odd number of people turn up for the game of if the captains agree that 2 weak players are equal to 1 strong player.

In the 7 rounds so far the scores have been:

ROUND1: TEAM 1 = AEHJK; TEAM 2 = BCDFIM; TEAM 1 SCORE = 10; TEAM 2 SCORE = 2; ROUND2: TEAM 1 = ABDFJ; TEAM 2 = CEGHI; TEAM 1 SCORE = 10; TEAM 2 SCORE = 3; ROUND3: TEAM 1 = ACEFK; TEAM 2 = BDGNO; TEAM 1 SCORE = 5; TEAM 2 SCORE = 1; ROUND4: TEAM 1 = ACFGI; TEAM 2 = BDJKPQ; TEAM 1 SCORE = 1; TEAM 2 SCORE = 4; ROUND5: TEAM 1 = ABIJ; TEAM 2 = CDFKOP; TEAM 1 SCORE = 3; TEAM 2 SCORE = 4; ROUND6: TEAM 1 = ABCFH; TEAM 2 = DEIKL; TEAM 1 SCORE = 10; TEAM 2 SCORE = 1; ROUND7: TEAM 1 = ACDIJ; TEAM 2 = BEFHK; TEAM 1 SCORE = 9; TEAM 2 SCORE = 0;

The organizers want to understand the strengths of each of the players so that they can organize more even match ups. If a teams strength is the sum of the strengths of each of the players then what is the strength of each player? It's possible that some combinations of players lead to a better result than would be predicted by a simple additive model. Is there a better way to accurately predict the results of the teams combinations? What is it?

• Is this a puzzle or an actual problem you face? Note that there's no reason you should be able to assign each player a number such that the sum of the numbers assigned to a team predicts how likely the team is to win against the other team. This kind of stuff depends strongly on the properties of the game. For example, in a game like Pokémon some players are weak or strong against others, and in particular there are loops where player A reliably beats player B, who reliably beats player C, who reliably beats player A. – Qiaochu Yuan Aug 15 '13 at 0:25
• There is probably some applicable voting system mathematics at work here, but I don't know what to call it. – rschwieb Aug 15 '13 at 0:45
• This is a real problem we face. Are there better tags to use? – user2027987 Aug 15 '13 at 1:02